Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T01:43:30.663Z Has data issue: false hasContentIssue false
  • English
  • Français

The Action Gambler and Equal-Sized Wagering

Part of: Game theory Stochastic processes

Published online by Cambridge University Press:  14 July 2016

David Hartvigsen
David Hartvigsen*
Affiliation:
University of Notre Dame
*
∗Postal address: 354 Mendoza College of Business, University of Notre Dame, Notre Dame, IN 46556-5646, USA. Email address: dhartvig@nd.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A gambler with an initial bankroll is faced with a finite sequence of identical and independent bets. For each bet, he may wager up to his current bankroll, and will win this amount with probability p or lose it with probability 1-p. His problem is to devise a wagering strategy that will maximize his final expected utility with the side condition that the total amount wagered (i.e. the total ‘action’) be at least his initial bankroll. Our main result is an expression that characterizes when the strategy of placing equal-sized wagers on all bets is optimal. In particular, for a given bankroll B, utility function f (concave, increasing, differentiable), and n bets, we show that it is optimal to wager b/n on each bet if and only if the probability of winning each bet is less than or equal to some value p⋆∈[½,1] (where p⋆ is an explicit function of B, f, and n). We prove the result by using a basic nonlinear programming technique.

Keywords

GamblingKelly criterionnonlinear programming

MSC classification

Secondary: 91A60: Probabilistic games; gambling 90C30: Nonlinear programming 60G40: Stopping times; optimal stopping problems; gambling theory 90C90: Applications of mathematical programming
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 35 - 54
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Bellman, R. and Kalaba, R. (1957). Dynamic programming and statistical communication theory. Proc. Nat. Acad. Sci. USA 43, 749–751.CrossRefGoogle ScholarPubMed
[2] Breiman, L. (1961). Optimal gambling systems for favorable games. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. I, University of California Press, Berkeley, pp. 65–78.Google Scholar
[3] Cetinkaya, S. and Parlar, M. (1997). Optimal nonmyopic gambling strategy for the generalized Kelly criterion. Naval Res. Logistics 44, 639–654.3.0.CO;2-D>CrossRefGoogle Scholar
[4] Chen, R. W., Shepp, L. A., Yao, Y.-C. and Zhang, C.-H. (2005). On optimality of bold play for primitive casinos in the presence of inflation. J. Appl. Prob. 42, 121–137.CrossRefGoogle Scholar
[5] DeGroot, M. H. (2004). Optimal Statistical Decisions. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
[6] Dubins, L. E. and Savage, L. J. (1976). Inequalities for Stochastic Processes (How to Gamble If You Must). Corrected republication of the 1965 edition. Dover, New York.Google Scholar
[7] Ethier, S. N. and Levin, D. A. (2005). On the fundamental theorem of card counting, with application to the game of trente et quarante. Adv. Appl. Prob. 37, 90–107.CrossRefGoogle Scholar
[8] Ethier, S. N. and Tavaré, S. (1983). The proportional bettor's return on investment. J. Appl. Prob. 20, 563–573.CrossRefGoogle Scholar
[9] Ferguson, T. S. (1965). Betting systems which minimize the probability of ruin. J. SIAM 13, 795–818.Google Scholar
[10] Finkelstein, M. and Whitley, R. (1981). Optimal strategies for repeated games. Adv. Appl. Prob. 13, 415–428.CrossRefGoogle Scholar
[11] Freedman, D. A. (1967). Timid play is optimal. Ann. Math. Statist. 38, 1281–1283.CrossRefGoogle Scholar
[12] Guu, S.-M. and Wang, S.-P. (2002). Optimal gambling strategy and relative risk aversion. J. Chinese Inst. Indust. Eng. 19, 34–40.Google Scholar
[13] Heath, D. C., Pruitt, W. E. and Sudderth, W. D. (1972). Subfair red-and-black with a limit. Proc. Amer. Math. Soc. 35, 555–560.CrossRefGoogle Scholar
[14] Karush, W. (1939). Minima of functions of several variables with inequalities as side conditions. , University of Chicago.Google Scholar
[15] Kelly, J. L. Jr. (1956). A new interpretation of information rate. Bell. System Tech. J. 35, 917–926.CrossRefGoogle Scholar
[16] Kuhn, H. W. and Tucker, A. W. (1961). Nonlinear programming. In Proc. 2nd Berkeley Symp. Math. Statist. Prob., ed. Neyman, J., University of California Press, Berkeley, pp. 481–492.Google Scholar
[17] Kulldorff, M. (1993). Optimal control of favorable games with a time limit. SIAM J. Control Optimization 31, 52–69.CrossRefGoogle Scholar
[18] Luenberger, D. G. (1998). Investment Science. Oxford University Press.Google Scholar
[19] Markowitz, H. M. (1952). Portfolio selection. J. Finance 7, 77–91.Google Scholar
[20] Molenaar, W. and van der Velde, E. A. (1967). How to survive a fixed number of fair bets. Ann. Math. Statist. 38, 1278–1280.CrossRefGoogle Scholar
[21] Pemberton, M. and Rau, N. (2001). Mathematics for Economists: An Introductory Textbook. Manchester University Press.Google Scholar
[22] Ross, S. M. (1974). Dynamic programming and gambling models. Adv. Appl. Prob. 6, 593–606.CrossRefGoogle Scholar
[23] Ruth, K. (1999). Favorable red and black on the integers with a minimum bet. J. Appl. Prob. 36, 837–851.CrossRefGoogle Scholar
[24] Samuelson, P. A. (1971). The ‘fallacy’ of maximizing the geometric mean in long sequences of investing or gambling. Proc. Nat. Acad. Sci. USA 68, 2493–2496.CrossRefGoogle ScholarPubMed
[25] Schweinsberg, J. (2005). Improving on bold play when the gambler is restricted. J. Appl. Prob. 42, 321–333.CrossRefGoogle Scholar
[26] Sun, W. and Yuan, Y.-X. (2006). Optimization Theory and Methods. Springer, New York.Google Scholar
[27] Research, Wolfram. (2008). Mathematica 6. Champaign, IL.Google Scholar
[28] Yao, Y.-C. (2007). On optimality of bold play for discounted Dubins–Savage gambling problems with limited playing times. J. Appl. Prob. 44, 212–225.CrossRefGoogle Scholar